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Economics 50: Intermediate Microeconomics
Summer 2010
Stanford University
Michael Bailey
Lecture 5: Production Functions and Profit Maximization
Overview
• The production function is the highest level of production the firm can produce with a set of inputs
• Most realistic production functions are monotonic, convex, and have diminishing returns
• When MP > AP, AP is increasing. When MP < AP, AP is decreasing
• In the short run, at least one factor of production is fixed
• The technical rate of substitution is the rate at which one input xj can be substituted for another
input xi to keep output constant
• A production function has
Constant Returns to Scale if f(λx) = λf(x)
Increasing Returns to Scale if f(λx) > λf(x)
Decreasing Returns to Scale if f(λx) < λf(x)
• The elasticity of substitution σ ≡ εKL ,TRSL,K
is a measure of the curvature of the isoquant and indicates
how substitutable inputs are for each other. A high σ indicates the inputs are good substitutes in the
production process
• A profit maximizing firm sets marginal revenue equal to marginal cost
• In a perfectly competitive market, the objective function of the profit maximizing firm is max pyf(x)−∑wixi and has a solution given by the optimality condition that the marginal revenue product of the
input equals the input price
• The factor demand functions, x∗i (py, w), are the optimal choice of the inputs as a function of the prices:
x∗i (py, w) = arg max pyf(x)−∑
wixi
1
• The supply function, y∗(py, w), is the optimal supply as a function of the prices:
y∗(py, w) = f(x∗1(py, w), ..., x∗n(py, w))
• The profit function, is the maximum profits attainable given prices:
Π(py, w) = pyy∗(py, w)−
∑wix
∗i (py, w)
• The weak axiom of profit maximization is that firm’s choices must yield higher profits than any other
point from its revealed production set
Producer Theory
So far, we have looked at the determinants of consumer demand. Individual demand is derived from con-
sumers’optimal choices and market demand is the summation of consumer demand. Now we will turn our
attention to the determinants of supply by analyzing the behavior of producers; we will see that many of the
theories and conclusions of consumer theory have an analog in producer theory. We assume that producers
are profit maximizers. Recall that profits are given by:
Profits = Revenue− Costs
= py − c(y)
Just as consumer theory looks at the implications of a model of utility maximizing consumers, producer
theory looks at the implications of a model of profit maximizing producers. There are two major constraints
on the producer: (1) technological constraints and (2) market constraints. Technological constraints dictate
how the ′y′ can be produced, or what inputs can be turned into the outputs. Market constraints dictate how
prices and costs are determined in the market. For example, in a perfectly competitive market the firm takes
the output price as given, whereas a monopolist sets the output price. We begin by assuming the firm is in
a perfectly competitive market that takes all output and input prices as given, and look at the technological
constraint side of profit maximization. Later in the course we will introduce different market constraints.
Technology
A technology is a way to combine inputs into outputs. For example, suppose that x = (x1, ..., xn) is a vector
of inputs and y = (y1, ..., yk) is a vector of outputs. A technology would be represented by some function
2
g(x) = y. The production set is the set of inputs and outputs such that the outputs are technologically
feasible to create with the inputs, or the set of inputs and outputs (x, y) such that there exists a technology
g() such that g(x) = y. We will assume in this class that the firm is only producing one output, y = y1.
Figure 1: A production set for a convex technology
Production Function
Because we assume the firm is a maximizer, we only are worried about those technologies that produce more
output, given the inputs, than any other technology. If the firm used a technology that produced less output,
then it would be losing potential revenue.
Definition 1 The production function is the maximum possible output given a vector of inputs, f(x) = y.
Example 2 Examples of production functions, often using labor (L) and capital (K) as inputs, include:
Perfect Complements f(L,K) = (min(aL, bK))γ
Perfect Substitutes f(L,K) = (aL+ bK)γ
Cobb Douglas f(L,K) = γLaKb
Constant Elasticity of Substitution (CES) f(L,K) = (aLα + bKα)γα
These have the same interpretation as in consumer theory. For example, perfect complements means the
firm will always use inputs in the same ratio aL = bK. γ is a term that we call a "technology shifter" because
it increases the amount of output the firm can get for the same amount of inputs.
3
The CES production function is the general form of the first three production functions depending upon
the value of α. When α = 1, it is equivalent to perfect substitutes. When α −→ 0, the production function
becomes Cobb-Douglas in the limit (take logs and use L’Hospitals Rule). When α −→ −∞, the production
function becomes perfect complements.
Definition 3 An isoquant is a level curve of the production function, or a set of inputs that will produce the
same amount of output y.
Isoquants are the analog to indifference curves in consumer theory. Note that in producer theory, the
level of production, y, is not arbitrary like utility, it has an actually physical meaning, 3 cars is not the same
thing as 6 cars. Production functions are not just a rank over bundles like a utility function, it literally tells
us how much of an output those inputs can produce given the production function. In consumer theory, we
always set γ = 1 because the actual level of utility doesn’t matter, in producer theory it does and we let γ
act as a parameter that dictates the level of production we can attain.
Example 4 f(L,K) = min(L,K)
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
L
K
Figure 2: Isoquants for the perfect complements production function f(L,K) = min(L,K) for productionlevels of 3 and 6
Example 5 Cobb Douglas f(L,K) = 2L1/3K2/3
4
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
10
L
K
Isoquants for the Cobb-Douglas production function f(L,K) = 2L1/3K2/3 with output levels equal to 5 and
10
Axioms of Technology
Axiom 6 A technology is monotonic if it can create at least as much output if an input is increased.
A production function is monotonic if ∂f(x)∂xi
≥ 0 for all inputs xi.
The monotonicity axiom just says that more inputs can only help the firm create more output, or at
worst not decrease output. Similar to consumer theory, monotonicity means that the isoquants cannot be
upward sloping and are not thick.
Axiom 7 A technology is convex if for A,B ∈ Production Set, then λA+ (1− λ)B ∈ Production Set.
If the technology is convex, the production function is concave: for two input vectors x1 and x2 that yield
the same output y, f(x1) = f(x2) = y, the production function is concave if f(λx1 + (1− λ)x2) ≥ y.
The convexity assumption means the firm can produce just as much output with the average of two input
bundles then with the bundles themselves. This means that the production function is concave, or always
lies above the line between two points, and the isoquants are convex.
5
Figure 3: This technology is convex: the production set is convex and the production function is concave
Figure 4: A technology that is not convex, the production function is not concave
6
Marginal Product and Average Product
Definition 8 The marginal product of factor xi,∂f(x)∂xi
≡ MPxi is the additional output from increasing
factor xi on the margin.
The marginal product is the analog to marginal utility, but note that the marginal product has an actual
meaning, the extra output is a physical amount, unlike marginal utility. Notice that the marginal product
is the slope of the production function.
Definition 9 The production function has diminishing returns in factor xi if∂2f(x)∂x2i
≤ 0.
Definition 10 The law of diminishing returns holds if limxi→∞∂2f(x)∂x2i
≤ 0 for all inputs xi; the marginal
product of all factors eventually are decreasing, holding all other factors constant.
The law of diminishing returns holds if eventually each additional input is less productive, holding all
other inputs constant. Think of a farm plot. The first worker is very productive and can work the best parts
of the land, whereas the next work is less productive and can still produce output, but not as much as the
first worker. As more and more workers are added, holding capital equipment and land fixed, each additional
worker becomes less productive being relegated to the marginal land and using the worst equipment. If the
law of diminishing returns did not hold, the world’s food supply could be grown in a flower pot.
Definition 11 The average product of factor xi,f(x)xi
= APxi , is the quantity of output produced per unit of
input xi.
The average product is the slope of the line from the origin to the point of interest ((xi, f(x)). If the
MPxi > APxi , then APxi must be increasing and if MPxi < APxi , then APxi must be decreasing. If you
add terms that are higher (lower) than average to the series, the average of the new series must be higher
(lower).
7
Figure 5: The average product is the slope of the line from the origin to the point on the production function.The marginal product is the slope of the line tangent to the production function
Figure 6: When MP > AP, AP is increasing. When MP < AP, AP is decreasing
8
Short Run versus Long Run
In the short run, at least one input (often referred to as a factor of production), is fixed at some level that
cannot be changed. In the long run, all inputs are variable. For example, a firm might have a fixed level
of capital corresponding to a factory that it has built. In the long run, it can produce more factories, or
demolish existing factories, so capital is completely unconstrained.
Example 12 Suppose that capital is fixed at K = K, then the production function f(L,K) = L1/2K1/2
is the short run production function. The long run production function is f(L,K) = L1/2K1/2. The fixed
variables are exogenous in the short run.
Technical Rate of Substitution
If the firm used one less unit of capital, K, how much more labor, L, would it need to employ so that
production remains the same? If we take the total derivative of the production function (with two inputs, L
and K), we would have that:
dy = MPLdL+MPKdK
To keep production constant we set dy = 0, and solve:
MPLdL+MPKdK = 0
MPLdL = −MPKdK
−dKdL
=MPLMPK
Thus to keep production constant, the rate at which we would substitute the two inputs to keep production
constant is MPLMPK
which is minus the slope of the isoquant−dKdL .We call this the technical rate of substitution1 :
Definition 13 The Technical Rate of Substitution of input xi for input xj , TRSxi,xj ≡MPxiMPxj
, is the rate at
which input xj can be substituted for input xi to keep output constant.
Notice that if we have∂TRSxi,xj
∂xi≤ 0, then the technology is convex.
Example 14 f(L,K) = 2L1/3K2/3
TRSL,K ≡MPLMPK
=2(1/3)L−2/3K2/3
2(2/3)L1/3K−1/3=
1
2
K
L
1Technically, this is the marginal technical rate of substitution, MTRS, which only considers infinitesimal changes. TheTechnical Rate of Substitution considers discrete changes −∆K
∆L.
9
Returns to Scale
Definition 15 A production function has
Constant Returns to Scale if f(λx) = λf(x)
Increasing Returns to Scale if f(λx) > λf(x)
Decreasing Returns to Scale if f(λx) < λf(x)
Notice that the definition of constant returns to scale is equivalent to homogeneity of degree 1. If the
production function is homogenous of degree r > 1, then it will satisfy increasing returns to scale, and if it
is homogenous of degree r < 1, it will satisfy decreasing returns to scale.
Remark 16 If a production function has increasing returns to scale, it will lie above the y = x1 line in the
one-input case (above the∑xi = y hyperplane in the multi-input case). Similarly, it will lie on or below the
y = x line if it has constant or decreasing returns to scale respectively.
Example 17 f(L,K) = ALaKb
f(λL, λK) = A(λL)a(λK)b
= λa+bALaKb
= λa+bf(L,K)
=⇒ homogenous degree r = a+ b
10
Figure 7: Three production functions that have increasing, constant, and decreasing returns to scale
Elasticity of Substitution
A measure of how substitutable two inputs are in a production process would be very useful. The technical
rate of substitution tells us how substitutable two inputs are at a certain point, but it does not inform us of
how substitutable the two goods are along the entire isoquant. If the firm varies the ratio of inputs greatly
for a change in the TRS between the inputs, then the inputs must be very substitutable for each other.
Conversely, if the TRS is changing, but the firm is not changing the input ratios, then the goods must be
complementary, otherwise they would profit from the changing TRS by changing the input ratio.
The curvature of the isoquant measures how much the input ratio changes for a change in the TRS. If
the isoquant is very curved, then the goods are very complementary because a large changes in the TRS
(slope of the isoquant) leads to small changes in the input ratio (slope of curve from origin to the point). If
the isoquant is nearly linear, small changes in the TRS would lead to large changes of the input ratio; the
inputs are very substitutable. One measure of the curvature of the isoquant, and thus the substitutability is
the elasticity of substitution.
Definition 18 The elasticity of substitution between inputs L and K, σ ≡ εKL ,TRSL,K
, is the percentage
change in the input ratio KL per percentage change in the TRSL,K .
The larger the elasticity of substitution, the more substitutable the inputs are in the production process.
11
Alternative ways of writing the elasticity of substitution are:
σ = εKL ,TRSL,K
=%∆K
L
%∆TRSL,K
=∂KL
∂TRSL,K
TRSL,KKL
=∂ ln(KL )
∂ lnTRSL,K
=1
∂TRSL,K∂KL
KL
TRSL,K
Example 19 Perfect Substitutes f(L,K) = aL+ bK
Because the isoquant is linear, the TRSL,K does not change along the curve, while the KL ratio does
change, so σ =∞.
Example 20 Perfect Complements f(L,K) = min(aL, bK)
This isoquant has a slope equal to infinity at the points where bK > aL, 0 when bK < aL, and undefined
when aL = bK. Consider a point where aL = bK, in a neighborhood around the point (for small changes
around the point), the TRS changes by an infinite amount (from 0 to ∞ or from ∞ to 0), whereas the ratio
of inputs KL changes very little, so σ = 0. At the other points, the TRS does not change in a neighborhood
around the point so σ =∞.
Example 21 f(L,K) = ALaKb
TRSL,K =K
L
a
b∂TRSL,K
∂KL=
a
b
KL
TRSL,K=
b
a
=⇒ εKL ,TRSL,K
=1
∂TRSL,K∂KL
KL
TRSL,K
= 1
12
Figure 8: σ measures how responsive KL is to a changing TRS
Profit Maximization
We can write the firm’s profits as a function of output, q:
Profits(q) = Revenues(q)− Costs(q)
The firm is a profit maximizer, so on the margin the additional profits from an additional unit of output
must be 0, otherwise it would produce more:
∂Profits(y)
∂y=∂Revenues(y)
∂y︸ ︷︷ ︸Marginal Revenue
− ∂Costs(y)
∂y︸ ︷︷ ︸Marginal Cost
= 0
This is equivalent to saying the firm would set marginal revenue equal to marginal cost. We will derive
this condition for various market conditions that change the marginal revenue and marginal cost.
Suppose the firm is in a perfectly competitive environment such that it takes the output price and all
input prices as given. It can buy as many inputs as it wants, and produce as much output as it wants,
without affecting the price (a firm that can alter the output price would be a monopolist or oligopolist, a
13
firm that can affect the input price is a monopsonist). In terms of notation:
y = output
x = (x1,..., xn) = vector of inputs
f(x) = y technology constraint (production function)
py = price of output
w = (w1, ..., wn) = vector of input prices
We can write the firms profits in the perfect competition case as:
profits = (py,−w) · (y, x)
= pyy − w1x1 − ...− wnxn
pyf(x)−∑
wixi
The objective function of the firm is:
maxx
pyf(x)−∑
wixi
which has the first order conditions:
py∂f(x)
∂xi− wi = 0 for all i
Which tells us that pyMPxi = wi at the optimum. We call pyMPxi the value of the marginal product, or
the marginal revenue product of xi (MRPxi). At the optimum, the value of the marginal product must be
equal to the input price for all inputs. If this were not true, then the firm could increase profits by using more
or less of that input. The second order condition (if there is one input x1) is that py∂2f(x)∂x21
≤ 0, or ∂2f(x)∂x21
≤ 0..
If the production function has diminishing returns in the input, then the optimum is a maximum.
Example 22 f(L,K) = LaKb, output price is py, and input prices are w and r for L and K respectively.
What values of L yield the highest profit?
14
max pyLaK
b − wL− rK
=⇒ apyLa−1K
b= w
=⇒ La−1 =w
apyKb
=⇒ L =
(w
apyKb
) 1a−1
= L∗(py, w, r)
Factor Demand, Supply, and Profit
The factor demand functions, x∗i (py, w), are the optimal choice of the inputs as a function of the prices:
x∗i (py, w) = arg max pyf(x)−∑
wixi
The supply function, y∗(py, w), is the optimal supply as a function of the prices:
y∗(py, w) = f(x∗1(py, w), ..., x∗n(py, w))
The profit function, is the maximum profits attainable given prices:
Π(py, w) = pyy∗(py, w)−
∑wix
∗i (py, w)
Example 23 f(L,K) = LaKb
We found the factor demand function(
w
apyKb
) 1a−1
= L∗(py, w, r). The supply and profit function are
given by:
y∗(py, w) = f(x∗1(py, w), ..., x∗n(py, w))
=
(w
apyKb
) aa−1
Kb
Π(py, w) = pyy∗(py, w)−
∑wix
∗i (py, w)
= py
(w
apyKb
) aa−1
Kb − w
(w
apyKb
) 1a−1
− rK
15
Graphical Solution
We can find the profit maximizing solution graphically in the one output, one input case. Let profits equal:
π = pyy − wx
=⇒ y =π
py+w
pyx
This is called an iso-profit line, a set of input combinations that yield the same profits. It has y-intercept
πpyand slope w
py. We can recast the profit maximization problem as "get to the highest iso-profit line given
y = f(x).” It is apparent that for an interior solution, the highest iso-profit line will be tangent to the
production function, otherwise there is an even higher iso-profit line that intersects the production function.
At the tangency, the slope of the iso-profit equals the slope of the production function, or MPx = wpy
=⇒
w = MRPx, which is the solution we found with calculus.
The maximizing condition w = MRPx does not necessarily hold at a corner solution. It could be the
highest attainable profits is 0 where no inputs are used. In the multiple input case, the solution might be
to use only one of the inputs, or to set some input to 0. We can amend our maximization problem to take
care of non-negativity constraints similar to the consumer demand problem. Given a monotone and concave
production set, if the factor demand is negative x∗i < 0 for some values of the prices, then set x∗i = 0 for
those values of the prices.
0 1 2 3 4 5 6 7 8 9 100
10
20
30
x
y
Figure 9: The profit maximizing firm will try to attain the highest iso-profit line given y = f(x). The highestiso-profit line is tangent to the production function at an interior solution.
16
0 1 2 3 4 5 6 7 8 9 100
10
20
30
L
y
Figure 10: The highest iso-profit line attainable has 0 profits (corner solution)
Examples of Profit Maximization
Example 24 f(L,K) =√L, prices (py, w)
maxL
py√L− wL
=⇒ py1
2L−1/2 = w
=⇒ L1/2 =py2w
=⇒ L∗(py, w) =( py
2w
)2
=⇒ y∗(py, w) =( py
2w
)=⇒ Π(py, w) = py
( py2w
)− w
( py2w
)2
=p2y
2w
Example 25 f(L,K) = 50L− 2L2, prices (py, w)
17
maxL
py(50L− 2L2
)− wL− rK
=⇒ 50py − 4pyL = w
=⇒ 4pyL = 50py − w
=⇒ L∗(py, w) = max
(0,
50
4− w
4py
)=⇒ y∗(py, w) =
( py2w
)=⇒ Π(py, w) = py
( py2w
)− w
( py2w
)2
=p2y
2w
The factor demand will be equal to 0 when 504 −
w4py≤ 0, or 50py ≤ w. We need to be careful about this
corner solution when calculating supply and profit:
y∗(py, w) = max
[0, 50
(50
4− w
4py
)− 2
(50
4− w
4py
)2]
Π(py, w) = max
{0, py
[50
(50
4− w
4py
)− 2
(50
4− w
4py
)2]− w
[50
4− w
4py
]}
Example 26 f(L,K) =√L+K, prices (py, w, r)
maxL,K
py√L+K − wL− rK
The FOCs for this problem yield: py 12 (L+K)−1/2 = w = r, which does not hold unless w = r. Notice that
if r > w, the firm will not want to use K as it contributes the same to production as L but is more expensive;
the two inputs are perfect substitutes. In that case, the firm sets K = 0, and resolves maxL,K py√L − wL
which is the same problem as (24). If w < r, then L = 0 and K has the same solution as (24). When w = r,
the firm is indifferent about using either input.
L∗(py, w, r),K∗(py, w, r) =
(( py2w
)2, 0)
if w < r(a( py
2w
)2, (1− a)
(py2r
)2)for some a ∈ [0, 1] if w = r(
0,(py
2r
)2)if w > r
18
y∗(py, w, r) =
( py2w
)if w < r( py
2w
)=(py
2r
)if w = r(py
2r
)if w > r
Π(py, w, r) =
p2y2w if w < r(
p2y2w
)=(p2y2r
)if w = r
p2y2r if w > r
Example 27 f(L,K) = L+K, prices (py, w, r)
maxL,K
py(L+K)− wL− rK
= L(py − w) +K(py − r)
Notice that if py > w, the firm can always increase profits by increasing L, and similarly if py > r, the
firm can always increase profits by increasing K. If the condition holds with <, the firm would always lose
profit and would never use that input. If the condition holds with =, the firm is indifferent between using
any amount of the input.
L∗(py, w, r) =
∞ if py > w
[0,∞) if py = w
0 if py < w
K∗(py, w, r) =
∞ if py > r
[0,∞) if py = r
0 if py < r
The supply will either be 0 or ∞ and the profit function will either be 0 or ∞.
An important result that this example demonstrates is that a production function with CRTS or IRTS
will have Π = 0 or ∞. To see this, suppose that the firm had CRTS had Π = pf(L,K)−wL− rK = a > 0.
If the producer scaled all inputs up by the same amount, the new profit would be:
pf(λL, λK)− wλL− rλK
= pλf(L,K)− wλL− rλK
= λ (pf(L,K)− wL− rK)
= λΠ
19
So the producer would set λ =∞ and earn infinite profits. This would only fail if Π = 0.
Example 28 f(L,K) = (LK)1/3, prices (py, w, r)
maxL,K
py(LK)1/3 − wL− rK =⇒
py1
3L−2/3K1/3 = w
py1
3L1/3L−2/3 = r
=⇒py
13L−2/3K1/3
py13L
1/3K−2/3=w
r
=⇒ K
L=w
r
=⇒ K =w
rL
=⇒ py1
3L−2/3
(wrL)1/3
= w subbing back into FOC
=⇒ L−1/3 =3w
py
(wr
)−1/3
=3
py
(rw2
)1/3=⇒ L =
(3
py
)−3 (rw2
)−1
=⇒ L∗(py, w, r) =(py
3
)3(
1
rw2
)=⇒ K∗(py, w, r) =
w
rL∗
=w
r
(py3
)3(
1
rw2
)=
(py3
)3(
1
r2w
)=⇒ y∗(py, w, r) = (L∗K∗)1/3
=
[(py3
)6(
1
r3w3
)]1/3
=(py
3
)2 1
rw
=⇒ Π(py, w, r) = pyy∗(py, w, r)− wL∗(py, w, r)− rK∗(py, w, r)
= py
(py3
)2 1
rw− w
(py3
)3(
1
rw2
)− r
(py3
)3(
1
r2w
)= 3
(py3
)3 1
rw− 2
(py3
)3(
1
rw
)=
(py3
)3 1
rw
20
Hotelling’s Lemma
Proposition 29 Hotelling’s Lemma
∂Π(py, w)
∂py= y∗(py, w)
∂Π(py, w)
∂w= −x∗i (py, w)
Hotelling’s Lemma is, like Shephard’s Lemma, an application of the envelope theorem. We expect a
change in py to affect the profit function, Π(py, w) = pyy∗(py, w)−
∑wix
∗i (py, w), in two ways: (1) directly
via ∂Π(py,w)∂py
(2) indirectly via ∂Π(py,w)∂y∗
∂y∗
∂pyand ∂Π(py,w)
∂x∗i
∂x∗i∂py
. The change in price will affect profits, but it will
also affect the supply and factor demands which in turn affect profits. But notice that the first derivative
in the indirect channels, ∂Π(py,w)∂y∗ and ∂Π(py,w)
∂x∗i, have to equal 0 since they are at the optimum. This is
the crux of the envelope theorem, when taking derivatives at optimal values, the indirect channels drop out.
Looking at Π(py, w) = pyy∗(py, w)−
∑wix
∗i (py, w), if we ignore how a change in py affects y∗ and x∗i it is
clear that ∂Π(py,w)∂py
= y∗(py, w) and ∂Π(py,w)∂w = −x∗i (py, w).
Example 30 f(L,K) = (LK)1/3, Π(py, w, r) =(py
3
)3 ( 1rw
)∂Π(py, w)
∂py= 3
(p2y
33
)(1
rw
)=(py
3
)2(
1
rw
)= y∗(py, w, r)
∂Π(py, w)
∂w= −
(py3
)3(
1
rw2
)= −L∗(py, w, r)
Revealed Profitability and the WAPM
Suppose we are given a firm’s input-output choices, is there any way we can tell whether the firm is a profit
maximizer? Assuming the firm’s technology is constant, the firm’s input-output choices reveal points on the
firm’s production function. If the firm is a profit maximizer, then the firm’s choices in each period must
yield higher profits then any other point on its production function. This is the basis for the weak axiom of
profit maximization (WAPM). Suppose that the firm chooses output yi = f(xi) in period i using inputs xi
when prices are (piy, wi) and thus makes profit πi = piyy
i − wixi (wixi =∑nj w
ijxij).
Definition 31 The firm’s input-output choices are consistent with the weak axiom of profit maximization if
its choices in each period are more profitable then its choices in all other periods.
πt = ptyyt − wtxt ≥ ptyys − wtxs for all periods t and s
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It is much easier to check for violations of WAPM (we say "WAPM" and not "the WAPM") than of
WARP because it is suffi cient to find one period where the firm could have made higher profits by choosing
another period’s input-output combination to find a violation. Intuitively, WAPM tells us that if the firm
could have made higher profits by choosing another point in its observed production set (all observed choices
are in the production set), then the firm is not a profit maximizer.
Figure 11: The firm’s choices are consistent with WAPM, profits in each period would be lower if the firmchose the other period’s input-output bundle
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Figure 12: This firm’s choices violate WAPM. In each period, the firm could make higher profits by choosingthe input-output bundle of the other period. If the firm is a profit maximizer, then technology must havechanged
Example 32 Suppose that we observe the firm’s choices (one output, one input) in three periods according
to the table below:
(yi, Li) (piy, wi)
Period 1 (2,1) (1,1)
Period 2 (5,4) (2,1)
Period 3 (10,8) (4,3)
We can calculate the profit in each period and how much profit the firm would have made having chosen
the other points on its observed production set:
Profit (y1, L1) (y2, L2) (y3, L3)
(p1y, w
1) π1 = p1yy
1 − w1L1 = $1 p1yy
2 − w1L2 = $1 p1yy
3 − w1L3 = $2
(p2y, w
2) p2yy
1 − w2L1 = $3 π2 = p2yy
2 − w2L2 = $6 p2yy
3 − w2L3 = $12
(p3y, w
3) p3yy
1 − w3L1 = $5 p3yy
2 − w3L2 = $8 π3 = p3yy
3 − w3L3 = $16
These choices are not consistent with WAPM because in each period the firm could have made higher
profits by choosing (y3, L3) = (10, 8).
Graphically, WAPM requires that the iso-profit line through each periods’choice must be higher than the
iso-profit line through all other periods’choices (holding prices constant for the period under consideration).
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Implications of WAPM
Suppose that we observe the firm for two periods: (yi, Li) for i = 1, 2, when prices are (piy, wi) for i = 1, 2.
WAPM requires:
π1 = p1yy
1 − w1x1 ≥ p1yy
2 − w1x2 (1)
π2 = p2yy
2 − w2x2 ≥ p2yy
1 − w2x1 (2)
Multiplying (2) by −1 yields:
w2x1 − p2yy
1 ≥ w2x2 − p2yy
2 (3)
Adding (1) and (3):
p1yy
1 − p2yy
1 + w2x1 − w1x1 ≥ p1yy
2 − p2yy
2 + w2x2 − w1x2
−∆py1 + ∆wx1 ≥ −∆py2 + ∆wx2
∆py2 −∆py1 + ∆wx1 −∆wx2 ≥ 0
∆p∆y −∆w∆x ≥ 0
where ∆p = (p2y − p1
y) and ∆w = (w2 − w1). If ∆w = 0, then ∆p∆y ≥ 0. This inequality implies that
the change in the output must be opposite in sign to the change in the output price. If the price of output
falls, output must increase (at least weakly). If ∆p = 0, then −∆w∆x ≥ 0, which implies that the change
in factor demand must be the same sign as the change in it’s price; if the wage increases, the demand for
labor must fall (at least weakly).
WAPM implies that the supply function must be upward sloping, and factor demands must be downward
sloping. It is surprising that such powerful results (that we struggled to find in consumer theory) fell out of
the simple principle of WAPM for 2 periods in such a parsimonious equation.
Properties of Factor Demand, Supply, and Profit
1. Factor demand functions are downward sloping
∂x∗i (py, w)
∂wi≤ 0
WAPM implies that −∆w∆x ≥ 0.
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2. The supply function is upward sloping∂y∗(py, w)
∂py≥ 0
WAPM implies that ∆p∆y ≥ 0.
3. Factor demand functions are homogenous of degree r = 0
x∗i (λpy, λw) = x∗i (py, w)
The slope of the iso-profit line does not change if all prices and wages change by the same amount (slope
of iso-profit is wpy
). We found earlier that the optimal input choice would yield a tangency between the
production function and the iso-profit line; if the slope of the iso-profit line does not change, then the
tangency condition does not change, and the optimal inputs do not change. The iso-profit lines will all
shift, however, so the profit level does change.
4. The supply function is homogenous of degree r = 0
y∗(λpy, λw) = y∗(py, w)
This falls out from factor demands being homogenous of degree r = 0 : y∗(λpy, λw) = f(x∗(λpy, λw)) =
f(x∗(py, w)) = y∗(py, w).
5. The profit function is increasing in the output price, and decreasing in input prices
∂Π(py, w)
∂py≥ 0
∂Π(py, w)
∂wi≤ 0
If this did not hold, then the firm must not be maximizing its profits as a profit-maximizing firm would
respond to an output price increase by increasing output (at least weakly), and an input price increase
by decreasing use of that input (at least weakly).
6. The profit function is homogenous of degree r = 1
Π(λpy, λw) = λΠ(py, w)
This falls right out from the condition that factor demands and the supply function are homogenous
of degree 0 : Π(λpy, λw) = λpyy∗(λpy, λw) −
∑λwx∗i (λpy, λw) = λ (pyy
∗(py, w)−∑wx∗i (py, w)) =
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λΠ(py, w). This is a very intuitive result, if all prices change by the same amount, all iso-profit lines will
have the same slope, but their y-intercepts πpy(i.e. level of profit) shift by a factor of λ; the iso-profit
line that is now tangent to the production function is the one that is consistent with profits equal to
λπ.
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